Continuous Functions Between Topological Spaces
Another, more abstract notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as in the case of metric spaces. A topological space is a set X together with a topology on X which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).
A function
between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image
is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology TX), but the continuity of f depends on the topologies used on X and Y.
This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.
An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions
to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.
Read more about this topic: Continuous Function
Famous quotes containing the words continuous, functions and/or spaces:
“The gap between ideals and actualities, between dreams and achievements, the gap that can spur strong men to increased exertions, but can break the spirit of others—this gap is the most conspicuous, continuous land mark in American history. It is conspicuous and continuous not because Americans achieve little, but because they dream grandly. The gap is a standing reproach to Americans; but it marks them off as a special and singularly admirable community among the world’s peoples.”
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“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
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“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)